Fundamentals of Plasma Physics

2.7 Sheath physics and Langmuir probe theory 55

to denote a potential measured relative to the plasma potential, i.e.,

φ ̄(x) =φ(x)−φplasma. (2.106)

The bias potential imposed on the probe (or wall) will be shielded out by the plasma within
a distance of the order of the Debye length;this region is the sheath. The relative potential
̄φ(x)varies within the sheath and has the two limiting behaviors:

lim

x→ 0

φ ̄(x) = φprobe−φplasma

lim

|x|>>λD

φ ̄(x) = 0. (2.107)

Inside the plasma, i.e., for|x|>>λD,it is assumed that the electron distribution function is
Maxwellian with temperatureTe.Since the distribution function depends only on constants
of the motion, the one-dimensional electron velocity distribution function must depend only
on the electron energymv^2 /2 +qeφ ̄(x), a constant of the motion, and so must be of the
form

fe(v,x) =

n 0

√

π 2 κTe/me

exp

(

−

(

mv^2 /2 +qeφ ̄(x)

κTe

))

(2.108)

in order to be Maxwellian whenx>>λD.
The electron density is

ne(x) =

∫∞

−∞

dvfe(v,0) =n 0 e−qe

̄φ(x)/κTe

. (2.109)

When the probe is biased negative with respect to the plasma, only those electrons with
sufficient energy to overcome the negative potential barrier will be collected by the probe.
The ion dynamics is not a mirror image of the electron dynamics. This is because a
repulsive potential prevents passage of particles having insufficient initial energy to climb
over a potential barrier whereas an attractive potential allows passage of all particles en-
tering a region of depressed potential. Particle density is reduced compared to the inlet
density for both repulsive and attractive potentials but for different reasons. As shown in
Eq.(2.109) a repulsive potential reduces the electron density exponentially (this is essen-
tially the Boltzmann analysis developed in the theory of Debye shielding). Suppose the
ions are cold and enter a region of attractive potential with velocityu 0 .Flux conservation
shows thatn 0 u 0 =ni(x)ui(x)and since the ions accelerate to higher velocity when falling
down the attractive potential, the ion density must also decrease. Thusthe electron den-
sity scales asexp(−

∣

∣qe ̄φ

∣

∣/κTe)and so decreases upon approaching the wall in response

to what is a repulsive potential for electrons whereas the ion density scales as 1 /ui(x)and
also decreases upon approaching the wall in response to what is an attractive potential for
ions.
Suppose the probe is biased negatively with respect to the plasma. Since quasi-neutrality
within the plasma mandates that the electric field must vanish inside the plasma, the po-
tential must have a downward slope on going from the plasma to the probe and the deriv-
ative of this slope must also be downward. This means that the potential ̄φmust have a
convex curvature and a negative second derivative as indicated in Fig.2.6. However, the